Delocalization of uniform graph homomorphisms from $\mathbb{Z}^2$ to $\mathbb{Z}$

Abstract

Graph homomorphisms from the Z d lattice to Z are functions on Z d whose gradients equal one in absolute value. These functions are the height functions corresponding to proper 3-colorings of Z d and, in two dimensions, corresponding to the 6-vertex model (square ice). We consider the uniform model, obtained by sampling uniformly such a graph homomorphism subject to boundary conditions. Our main result is that the model delocalizes in two dimensions, having no translation-invariant Gibbs measures. Additional results are obtained in higher dimensions and include the fact that every Gibbs measure which is ergodic under even translations is extremal and that these Gibbs measures are stochastically ordered.

Three lectures on random proper colorings of $\mathbb{Z}^d$

“…then the distribution of h L converges locally as L → ∞ to a Z 2 even -translation-invariant Gibbs measure (an analogous statement holds in any dimension [21,Theorem 1.1]). Thus, Theorem 2.2 is a consequence of the following statement.…”

“…In this section we discuss the main ideas in the proof of Theorem 2.6 following [21,Theorem 3.1]. The results in Section 2.5.1 and Section 2.5.2 hold in any dimension d ≥ 1 while the results of Section 2.5.3 are restricted to dimension d = 2.…”

“…In every dimension d: Every Z d even -ergodic Gibbs measure is extremal. We do not provide a proof of the lemma here (see [21,Section 5]) and content ourselves with a description of the main steps.…”

The concentration inequality for a discrete height function model perturbed by random potential

“…[PSY13,Pel17,LT20]) and graph homomorphisms from certain discrete graphs to Z (e.g. [BHM00,Kah01,Gal03,CPST20]).…”

Deducing a variational principle with minimal a priori assumptions

“…A microscopic state or a height function is a graph homomorphism h Rn : R n → Z for some n, and a macroscopic state or asymptotic height function is a Lipschitz function h R : R → R. In the two-dimensional case, this model is equivalent to the six-vertex model with uniform weights (cf. [vB77,CPST18]). Full details of the model under study are given in Section 2.…”